Relativistic quantum mechanics of the Majorana particle: quaternions, paired plane waves, and orthogonal representations of the Poincar\'e group

The standard momentum operator $-i \nabla$ can not be accepted as observable in relativistic quantum mechanics of the Majorana particle. Instead, one can use axial momentum operator recently proposed in Phys. Lett. A {\bf383}, 1242 (2019). In the present paper we report several new results related to the axial momentum which elucidate its usability. First, a new motivation for the axial momentum is given, and the Heisenberg uncertainty relation checked. Next, we show that the general solution of time evolution equation in the axial momentum basis has a connection with quaternions. Single traveling plane waves are not possible in the massive case, but there exist solutions which consist of asymmetric pair of plane waves traveling in opposite directions. Finally, pertinent real orthogonal and irreducible representation of the Poincar\'e group -- consistent with the lack of antiparticle -- is unveiled.


Introduction
The discovery of non vanishing mass of neutrinos has led to many conjectures about the nature of these particles. In particular, it is possible that they are relativistic massive fermions of the Majorana type. While state of the art description of fundamental particles is provided by quantum field theory (notwithstanding its well-known problems), the slightly older framework of relativistic quantum mechanics also is useful, especially in the case of single particle. Relativistic quantum mechanics has many important applications in such branches of contemporary physics as atomic physics, theory of elementary particles, and even condensed matter physics, see, e.g., [1], [2] and [3]. Among relativistic wave equations, the most popular is of course the one proposed by P. A. M. Dirac, but other equations are interesting as well, in particular the Proca and the Salpeter equations recently discussed in, respectively, [4] and [5]. There is no doubt that relativistic quantum mechanics is the source of important insights. Quantum mechanics of the Majorana particle is not exception to this.
Relativistic quantum mechanics of a free Majorana particle significantly differs from quantum mechanics of the Dirac particle, and it has several unexpected features. Certain aspects of it have already been considered in [6], [7], [8], [9] and [10]. There is an interesting problem concerning momentum observable, to the best of our knowledge first considered in [7], and recently readdressed in [10] where the axial momentum operator has been proposed as the momentum observable. In the present paper we apply expansions into eigenfunctions of the axial momentum operator in order to discuss general solution of the time evolution equation as well as the relativistic invariance.
The time evolution equation for the Majorana particle coincides with the Dirac equation 1 , in which certain Majorana-type representation for Dirac matrices γ µ is assumed, that is all these matrices are purely imaginary. The crucial difference with the Dirac particle is that all four components of the bispinor ψ are real numbers. This is consistent with Eq. (1) because the matrices γ µ are imaginary, and m is a nonnegative real number. In consequence, the relevant Hilbert space H consists of all real normalizable bispinors, and the pertinent algebraic number field is that of real numbers R, instead of the more common in quantum mechanics algebraic field of complex numbers C. Let us note that there are other formulations of Majorana quantum mechanics, see, e.g., [6] and [9], but they are equivalent to the one adopted here. Quantum mechanics with real numbers or even quaternions in place of the algebraic field of complex numbers is not very popular, but it has been thoroughly discussed in literature, see for example [11], [12] and [13]. In particular, it is known that in the real and quaternionic cases the discrete symmetries P , T , and C are represented by unitary operators, while in the complex quantum mechanics also anti-unitary symmetry operators can appear.
The standard momentum operatorp = −i∇ turns real bispinors into imaginary ones, therefore it is not operator in H. New momentum-like operator is needed. Such operator -called axial momentum and denotedp 5 -has been proposed in [10], namelyp 5 = −iγ 5 ∇ in the Schroedinger picture. It is Hermitean, and its spectrum is continuous. Moreover, it can be regarded as the generator of spatial translations. On the other hand, it has certain rather peculiar features. First, it does not commute with Hamiltonian in the case of massive Majorana particle (m > 0) . In consequence, its direction is not constant in time in the Heisenberg picture -the axial momentum contains a rotating component of the magnitude m/E p , where E p = m 2 + p 2 5 is the energy of the particle [10]. This component is negligibly small at high energies, but it can not be neglected at the energies comparable with m. The eigenfunctions of the axial momentum can be used as a basis for Fourier-type expansion of time-dependent wave functions of the Majorana particle [10]. It turns out that in place of simple time-dependent U(1) phase factors known from the case of Dirac particle there are cumbersome time-dependent SO(4) matrices.
In the present paper we continue the investigations initiated in [10]. We begin with a new motivation for the axial momentum operator. It is based on a mapping between the Majorana and Weyl bispinors. We show that the classic position vs. momentum uncertainty relation remains unchanged when the standard momentum operator is replaced by the axial one. Next, we examine the general solution of the wave equation expanded in the basis of eigenfunctions of the axial momentum ψ p (x). It turns out that it can be regarded as position and time dependent quaternion. Next, we transform the solution to a more convenient form which does not contain the SO(4) matrices, namely we rewrite it as a superposition of traveling plane waves, see formula (13) below. That this is at all possible is a surprise because the direction of the axial momentum is not constant in time if m > 0. Interestingly, it turns out that in the massive case the plane waves nec-essarily come in pairs. The paired plane waves have the opposite wave vectors p and −p, hence they travel in the opposite directions. Their amplitudes are not equal, the ratio is 1 : m/E q .
Finally, we elaborate on the Poincaré invariance of the model using the amplitudes introduced in the basis of eigenfunctions of the axial momentum. We find rather straightforward realization of the relativistic invariance in the space of these amplitudes, which is rather encouraging result. The obtained representation of the Poincaré group in the massive case is orthogonal, irreducible, and equivalent to real version of the well-known spin 1/2 unitary irreducible representation. Recall that in the case of Dirac particle one obtains a reducible representation composed of two spin 1/2 irreducible representations. Such representations discussed in the framework of relativistic quantum mechanics usually reappear unchanged when one considers single particle sectors for the related quantum field. In the Dirac case the two spin 1/2 representations correspond to the particle and its antiparticle. In the Majorana case we expect the particle only.
Our overall conclusion is that the axial momentum is a reasonable replacement for the ordinary momentum (which should not be used in the Majorana case anyway). Certain peculiar features present in the case of massive Majorana particle, like the discussed in Section 3 mixing of modes with opposite axial momenta, are negligible at energies E p ≫ m. At lower energies however they can not be neglected. We regard them as intrinsic physical features of the relativistic massive Majorana particle.
The paper is organized as follows. In the next section we introduce the axial momentum using the mapping between the Weyl and Majorana bispinors and we derive the uncertainty relation. In Section 3, after a brief recap of necessary results from [10], we point out the connection with quaternions, and we discuss the traveling plane waves. Section 4 is devoted to analysis of the representation of the Poincaré group that exists in the space of solutions of the evolution equation.
2 The axial momentum: new motivation, and the Heisenberg uncertainty relation Throughout this paper we work with the Dirac matrices γ µ in a Majorana-type representation, i.e., the matrices are purely imaginary. Then also the matrix γ 5 = iγ 0 γ 1 γ 2 γ 3 is purely imaginary. Furthermore, γ 5 is Hermitian, hence also antisymmetric: γ T 5 = −γ 5 , and γ 2 5 = I , where I is the 4 by 4 unit matrix. We work with the following set of the Dirac matrices Here σ k are the Pauli matrices, and σ 0 is the 2 by 2 unit matrix. In all Majorana-type representations charge conjugation C is represented just by the complex conjugation. Therefore, the Majorana bispinors, which by definition are invariant under C, have only real components in such representations. The operatorp 5 commutes with C, in contradistinction top.
The motivation for the axial momentum given in [10] refers to a Lagrangian in classical field theory and to the Noether theorem. Moreover, it applies to the massless case (m = 0) only. Below we give an independent, simple and more general motivation.
There is a simple one-to-one mapping M between linear spaces of the Majorana bispinors and right-handed (or left-handed) Weyl bispinors. From arbitrary right-handed Weyl bispinor φ, which by definition has the property γ 5 φ = φ, we form ψ = φ + φ * ≡ M(φ), which is real, hence Majorana, bispinor. The asterisk denotes the complex conjugation. Now, because the matrix γ 5 is purely imaginary, φ * is a left-handed bispinor, γ 5 φ * = −φ * . It follows that γ 5 ψ = φ − φ * , and φ = (I + γ 5 )ψ/2 ≡ ψ R , φ * = (I − γ 5 )ψ/2 ≡ ψ L . This shows that the mapping M is invertible. Notice that it preserves linear combinations only if their coefficients are real. The Weyl bispinors are complex, hence the standard momentum operator p = −i∇ is well-defined for them. In particular, it commutes with the γ 5 matrix, therefore alsopφ is right-handed Weyl bispinor. Let us find the Majorana bispinor that corresponds topφ: We see that the standard momentum operator in the space of right-handed Weyl bispinors gives rise to the axial momentum operator in the space of Majorana bispinors.
The axial momentum commutes with γ 5 , therefore it can be used also in the space of right-handed Weyl bispinors. However, in this space it coincides withp because The presence of the one-to-one mapping M might suggest that the two quantum mechanics, Majorana and Weyl, are equivalent to each other. For the equivalence, the mapping M should preserve scalar product. It turns out that it is not the case. Let us take ψ 1 = M(φ 1 ), ψ 2 = M(φ 2 ), and compare the scalar product of the Majorana bispinors d 3 x ψ T 1 ψ 2 with the scalar product d 3 x φ † 1 φ 2 of the corresponding Weyl bispinors. We have Here we have used the identity φ T 1 φ 2 ≡ 0, which follows from the antisymmetry of γ 5 : Thus we see that in general the scalar product is not preserved by M, Note also the differences in evolution equations. In the Weyl case, the evolution equation has the form (1) with m = 0, namely iγ µ ∂ µ φ = 0, while in the Majorana case m = 0 is allowed. Using the mapping inverse to M one can of course transform Eq. (1) for the Majorana bispinor ψ to the space of right-handed Weyl bispinors -we obtain iγ µ ∂ µ φ − mφ * = 0, which is known as the Dirac equation for φ with the Majorana mass term (recall that φ * is the charge conjugation of φ). This last equation can not be accepted as a quantum mechanical evolution equation for the Weyl bispinor φ because it is not linear over C -it is linear only over R. The point is that the Hilbert space of the right-handed Weyl bispinors 2 is linear over C, therefore also quantum mechanical evolution equation for these bispinors should be linear over C, otherwise the superposition principle is broken. Commutator of the axial momentum with position operatorx has the form which differs by γ 5 from the commutator [x j ,p k ]. In spite of the difference, the implied uncertainty relation has the usual form The uncertainty relation is obtained in the standard manner. Let us consider where α is a real variable. It is clear that I(α) ≥ 0. On the other hand, using commutator (2) we have We know that this quadratic polynomial in α does not have two distinct real roots. The uncertainty relation follows as the necessary and sufficient condition for this.
In the massive case (m > 0) the axial momentum has nontrivial time evolution in the Heisenberg picture, because it does not commute with the Hamiltonianĥ shown below. This aspect is discussed in detail in [10].

Time evolution of the axial momentum amplitudes and quaternions
The general solution of Eq. (1) in the basis of eigenfunctions of the axial momentum was found in [10]. It is complete from theoretical viewpoint, but rather clumsy if one thinks about concrete applications. Below we transform that solution to a much simpler form. Furthermore, we point out that the general solution can be described in terms of quaternions. Such a link of the Majorana quantum mechanics with the algebra of quaternions is yet another intriguing feature of it, in addition to the non Hermitian Hamiltonianĥ and non conservation of the axial momentum in the case of free massive Majorana particle. Let us begin by recalling necessary facts from the paper [10]. The Dirac equation (1) is rewritten as where the Hamiltonianĥ is real and anti-symmetric, but it is not Hermitian if m = 0. Nevertheless, the scalar product The time evolution operator is orthogonal one.
The normalized eigenfunctions of the axial momentum have the form They obey the conditionŝ where v is an arbitrary constant, normalized (v T v = 1) and real bispinor. We call the functions ψ p (x) the axial plane waves 3 . Note that exp(iγ 5 px) = cos(px)I + iγ 5 sin(px).
The expansion of ψ(x, t) into the axial plane waves has the form where the basis bispinors v (±) α obey the conditions The eigenvalues ±|p| correspond to helicities ±1/2 , respectively, [10]. They are double degenerate (α = 1, 2). Thus each single mode in (5) is common normalized eigenstate ofp 5 and of the helicity. The index α = 1, 2 reflects the degeneracy of the common eigenstates which is an artefact of the reality of our Hilbert space.
The basis bispinors have the following form v (+) They are orthonormal, where ǫ, ǫ ′ = +, − refer to the helicity, and α, β = 1, 2. The basis (6) has quite remarkable properties: it is real; generated from v Time dependence of the real amplitudes c α (p, t), d α (p, t) in expansion (5) is found by solving Eq. (3). To this end, the amplitudes are split into the even and odd parts, , t), and analogously for d ′ , d ′′ . Furthermore, we introduce the notation and E p = m 2 + p 2 . The time dependence of the amplitudes is given by following formulas [10] c(p, t) = exp(tE p K + (p)) c(p, 0), d(p, t) = exp(tE p K − (p)) d(p, 0). (7) The matrices K ± (p) are anti-symmetric, hence the matrices exp(tE p K ± (p)) belong to the SO(4) group. Because K 2 ± = −I, we have the formula exp(tE p K ± (p)) = cos(tE p )I + sin(tE p )K ± (p).
Here we end the recapitulation of the relevant for this work facts from [10].
Using expansion (5) we immediately obtain the Plancherel formula where the amplitudes c α correspond to ψ 1 , and c α to ψ 2 . Let us remind that the integration variable p is the eigenvalue of the axial, not the ordinary, momentum. Splitting the amplitudes into the even and odd parts, we may write Because the time evolution is given by the orthogonal matrices, as shown in formulas (7), we again see that the scalar product is constant in time.
In the general solution of Eq. (3) quoted above the amplitudes c α , d α are split into the even and odd components which mix during the time evolution, see formulas (7). It turns out that the solution can be rewritten in a more transparent form. To this end, we use formulas (7) and (8), as well as the concrete form (6) of the basis bispinors. After straightforward and somewhat lengthy calculation the general solution is transformed to the following form In formula (10) we have the initial values of the full amplitudes c α , d α , while in (7) the even and odd parts appear separately. Notice that Intriguingly, the general solution (7) and its equivalent form (10) can be rewritten in terms of quaternions. The quaternionic unitsî,ĵ,k are introduced as follows:î = iγ 5 ,ĵ = iγ 0 ,k = −γ 5 γ 0 = iγ 1 γ 2 γ 3 .
The bispinor basis v (±) α (p) is generated from v (+) 1 (p) by acting withî,ĵ,k, see formulas (6). Moreover, all matrices present in formulas (7) and (10) can be expressed by I,î,ĵ,k. In particular, K ± (p) = ∓n 2î ± n 3ĵ + n 1k . Therefore, the time evolution of the amplitudes c, d at each fixed value of the axial momentum p is given by a time dependent quaternion.
Solution (10) can be written in a Fourier form, in which no matrices are present, only trigonometric functions and the basis bispinors (6). This is possible because the quaternions acting on the basis bispinors do not yield any new bispinors, but only permute them. This form of solution (10) reads where the V .. stand for linear combinations of the basis bispinors, namely

Formula (12) is a convenient starting point for analysis of concrete examples of solutions.
Solution (12) is a superposition of standing plane waves. In order to rewrite it in terms of traveling plane waves we use trigonometric formulas such as cos(px) cos(E p t) = 1 2 (cos(px − E p t) + cos(px + E p t)), etc. We obtain In these formulas p ≡ |p|, E p = p 2 + m 2 , and the amplitudes c 1 , c 2 , d 1 , d 2 are the ones present in formula (10) (the arguments (p, 0) have been omitted for brevity). Let us remind again that p is the eigenvalue of the axial momentum. Let us consider now a single mode with fixed value q of the axial momentum, i.e., we put in the formulas above c α (p, 0) = c α δ(p − q), d α (p, 0) = d α δ(p − q), where c α , d α , α = 1, 2, are constants now. In the massless case, We see that in this case the A + , B + part on the r.h.s. of formula (13) is independent of the A − , B − part. In particular, we can put one of them to zero in order to obtain a plane wave propagating in the direction of q or −q.
The massive case is different -the plane wave always has the two components propagating in the opposite directions, q and −q. If we assume that A − = 0, simple calculation shows that also Let us put the constants d 1 = d 2 = 0. In the massless case this assumption gives the plane wave moving in the direction q, In the massive case all four components in (13) do not vanish. However, the amplitudes of the −q components, i.e., A − , B − , are negligibly small in the high frequency limit, i.e., when m/E q ≪ 1. In this limit On the other hand, in the limit of long waves, i.e., q ≪ m, and In this case the q and −q components have approximately equal magnitudes.

Relation with irreducible representations of the Poincaré group
The Poincaré transformations of the real bispinor ψ(x) have the standard form, with S(L) = exp(ω µν [γ µ , γ ν ]/8), where ω µν = −ω νµ parameterize the proper orthochronous Lorentz group, L = exp(ω µ ν ), in a vicinity of the unit element. Below we show that in the massive case these transformations imply transformations of the axial momentum dependent amplitudes which coincide with the real form of a single standard unitary Wigner's representation with spin 1/2. This representation being real and unitary is in fact orthogonal one. The conclusion is that as far as the relativistic transformations is the issue, the expansion into the axial plane waves of the Majorana bispinor has the properties expected for a single massive spin 1/2 particle.
We do not discuss here representations of the Poincaré group pertaining to the massless Majorana particle (m = 0). There are two independent irreducible representations with the helicities ±1/2. This case is simpler because now the operatorp 5 commutes with the Hamiltonian. It will be presented in pedagogical notes [14].
We start from the following expansion into the eigenstates of the axial momentum where v(p, t) is a real bispinor, and E p = m 2 + p 2 . Equation (3) gives time The reason for v(−p, t) in the last term on the r.h.s. is that γ 0 anti-commutes with γ 5 and therefore γ 0 exp(iγ 5 px) = exp(−iγ 5 px)γ 0 . Taking time derivative of Eq.

Let us write its general solution in the form
where the argument of v − is −p for later convenience. Then ψ(x, t) can be written as where in the last term we have changed the integration variable to −p. Furthermore, Eq. (16) is satisfied by v(p, t) of the form (17) only if v ± (p) obey the following conditions Applying the transformation law (14) with a = 0 to solution (18), we obtain Lorentz transformation of the bispinors v ± (p), where now we use the four-vector p instead of p for convenience in notation: v + (p) ≡ v + (p) and p 0 = E p . The spacetime translations x ′ = x + a are represented by SO (4) factor v ′ ± (p) = e ±iγ 5 pa v ± (p).
In the massive case, v − (p) can be expressed by v + (p), see (19). The scalar product ψ 1 |ψ 2 = d 3 x ψ T 1 (x, t)ψ 2 (x, t) acquires explicitly Poincaré invariant (and time independent) form where v 1+ (p) = v T 1+ (p)γ 0 , and v 1+ (v 2+ ) corresponds to ψ 1 (ψ 2 ) by formula (18). Transformations (20), (21) are unitary with respect to this scalar product. Thus, we have here real unitary, i.e., orthogonal, representation of the Poincaré group. It turns out that it is irreducible and equivalent to a real version of the standard spin 1/2 unitary representation. Detailed analysis of the representation is given below. Let us recall that in the case of massive Dirac particle one finds a reducible representation which is a direct sum of two spin 1/2 representations.
Representation (20) can be cast in the standard form which involves the Wigner rotations and a representation of SU(2) group [15]. To this end, we introduce the standard momentum where i = 1, 2, 3, 4, and v i ( p ) = δ ik (the factor m is included for dimensional reason). Here again we use the four momentum in the notation as in (20).
Let us find the relativistic transformation law of the amplitudes a k (p). In the case of Lorentz transformations, using (20 The Lorentz transformation R(L, p)) = H −1 (p)LH(L −1 p) leaves (0) p invariantit is a rotation, known as the Wigner rotation. Therefore, we may write The invariance of the scalar product (24) implies orthogonality of the 4 by 4 real matrix with the elements D ki (R(L, p)). The space-time translation ψ ′ (x) = ψ(x − a) results in a change of the amplitudes 4 , a i (p) → a i (p). Let us compute a i (p). Using formula (21) we have v ′ + (p) = a k (p)v k (p) = e iγ 5 pa a k (p)v k (p) = a k (p)S(H(p))e iγ 5 pa v k ( At this point it is convenient to choose the basis v k ( p ) is equal to δ ik / √ m. In this basis In consequence, a l (p) = (e iγ 5 pa ) lk a k (p).
The matrix e iγ 5 pa is orthogonal, and the scalar product (24) is of course invariant with respect to the transformations (27).
Formula (25) opens the way to identification of the pertinent orthogonal representation of the Poincaré group. This representation is uniquely characterized by representation (26) of the Wigner rotations [15]. In order to identify this last representation it suffices to take in formula (25) p = The matrices S(R) have the form They form a subgroup of the SO(4) group. There exist real orthogonal matrices O such that whereî,ĵ,k are the quaternions introduced in the previous section. For example, one may take the matrix Thus, the matrices OS(R)O −1 are elements of the algebra of quaternions, and as such they can be written in the form where s 0 , s k are real functions of the parameters ω ik . These matrices also belong to the SO(4) group. Furthermore, because On the other hand, let us consider the spin 1/2 representation T (u) of SU(2) group, T (u)ξ = uξ, where u ∈ SU(2) and ξ is a two-component spinor (in general complex). This representation can be rewritten in real form simply by using the real and imaginary parts. Thus we write and αα * +ββ * = (α ′ ) 2 +(α ′′ ) 2 +(β ′ ) 2 +(β ′′ ) 2 = 1. Next, we form the four-component real vector ξ and the 4 by 4 real matrixT (u): It turns out that ξ u =T (u) ξ, where ξ u ≡ T (u)ξ. The matrixT (u) can be rewritten in terms of the quaternions, The r.h.s. of this formula coincides with the r.h.s. of formula (28) if α ′ = s 0 , β ′ = s 1 , β ′′ = s 2 , and α ′′ = s 3 .
In conclusion, the representation of the Wigner rotations given by the matrices S(R) is equivalent to the real form of the spin 1/2 representation T (u) of SU(2) group. Thus, we have found that the representation of the Poincaré group is the spin 1/2, m > 0 representation. Notice that we have obtained just one such spin 1/2 representation. In the case of Dirac particle a direct sum of two spin 1/2 representations appears, one for particle and the other for antiparticle.

Summary and remarks
1. Let us summarize our main results. We have shown that the axial momentum operator for the Majorana particle is related to the ordinary momentum for the Weyl particle by the one-to-one mapping between the two models, and that it obeys the Heisenberg uncertainty relation. Next, using the eigenfunctions and eigenvalues of the axial momentum operator, we have written the general solution of the Dirac equation for the real bispinor in the form of superposition of traveling plane waves, with the eigenvalues p of the axial momentum playing the role of wave vectors, i.e., giving the wave length and the direction of propagation. In the massive case this superposition has the special feature that the plane waves come in pairs with the opposite axial momenta, p and −p This is a consequence of the fact that in the massive case the axial momentum does not commute with the Hamltonianĥ . Therefore, the eigenvectors ofp 5 are not stationary states -the minimal stationary subspace in the Hilbert space is spanned by the two modes p, −p. The presence of such paired plane waves could perhaps serve as a signature of the massive Majorana particle. This effect is relatively small at high energies, but quite sizable at energies close to the rest mass of the particle. Finally, we have shown that using the axial momentum basis one can unveil the pertinent irreducible spin 1/2 representation of the Poincaré group.
Apart from the results listed above, there are quite interesting purely theoretical aspects, namely the reformulation in terms of quaternions, and fully-fledged relativistic quantum mechanics over the algebraic field of real numbers R in place of complex numbers.
We conclude that the axial momentump 5 = −iγ 5 ∇ can be accepted as the replacement for the ordinary momentump = −i∇. This latter operator is not an observable for the Majorana particle because it does not commute with the charge conjugation C, in contradistinction top 5 . In fact, we think that the axial momentum is the proper observable to be used in theoretical analysis of experimental data for relativistic Majorana particles, when they are available.
2. The investigations of the axial momentum can be continued in several directions. In our opinion, two are especially interesting. First, we would like to check time evolution of wave packets with certain fixed initial profile of the axial momentum. Formulas (12) and (13) seem to be a good starting point for work in this direction. We believe that such a basic knowledge about evolution of wave functions more general than the plane waves can facilitate searches for the Majorana particles.
The second very interesting topic is the application of the axial plane waves in quantum theory of the Majorana field. The amplitudes a i (p) introduced right above formula (24), where p = (p 0 , p) with p being eigenvalue of the axial momentum and p 0 = m 2 + p 2 , have clear transformation law with respect to the Poincaré group. This fact suggests that precisely these amplitudes should be replaced by creation and annihilation operators of the Majorana particle when quantizing the Majorana field.
Finally, one may apply the axial momentum instead of the ordinary momentum in quantum mechanics of the Dirac particle. Here the ordinary momentum has the advantage -it commutes with the Dirac Hamiltonian in the case of free particle -but the use of the axial momentum, which is after all a legitimate observable, can lead to new insights.